Complex wavenumber and circulating wave
We introduce a distance attenuation due to the air damping. In the description of wave motion in the viscous medium, it is convenient to introduce a complex wavenumber $\tilde{k}$. Moreover, we consider the circulating wave. The sound pressure is superposed at the arbitrary positions to each other, it is possible to express the pressure of the clockwise moving wave $p_{cw}(x,t)$ and counterclockwise moving wave $p_{ccw}(x-L_c,t)$ in consideration of the circulating wave as Eq.(20). So, sound pressure distributions around the tire cavity resonance can be expressed by the one-dimensional wave equation in consideration of circulating wave and air damping.
\begin{eqnarray}
p(x,t) & = & p_{cw}(x,t)+p_{ccw}(x-L_c,t) \nonumber \\
& = & p_0\exp(j\omega t)\sum_{i=0}^{n}\bigl[\exp\{-j\tilde{k}(x+iL_c)\}+\exp[j\tilde{k}\{x-(i+1)L_c\}]\bigr] \tag{20}
\end{eqnarray}
